Investigation of Thermal Adiabatic Boundary Condition on Semitransparent Wall in Combined Radiation and Natural Convection
G Chanakya, Pradeep Kumar Numerical Experiment Laboratory (Radiation and Fluid Flow Physics) School of Engineering Indian Institute of Technology Mandi Mandi, Himachal Pradesh, India 175075
Abstract
Two thermal adiabatic boundary conditions arise on the semitransparent win-
dow owing to the fact that whether semitransparent window allows the energy
to leave the system by radiation mode of heat transfer. It is assumed that
being low conductivity of semitransparent material, energy does not leave by
conduction mode of heat transfer. This does mean that the semitransparent
window may behave as only conductively adiabatic (qc = 0) or combinedly con-
ductively and radiatively adiabatic (qc +qr = 0). In the present work, the above two thermal adiabatic boundary conditions have been investigated in natural
convection problem for the Rayleigh number (Ra) 105 and Prandtl number(Pr)
0.71 in a cavity, whose left vertical wall has been divided into upper and lower
parts in the ratio of 4:6. The upper section is semitransparent window, while
lower section is isothermal wall at a temperature of 296K. A collimated beam is
irradiated with different values (0, 100, 500 and 1000 W/m2) on the semitrans-
parent window at an angle of 450. The cavity is heated from the bottom by arXiv:2007.12484v1 [physics.flu-dyn] 23 Jul 2020 convective heating with free stream temperature of 305K and heat transfer coef-
Email address: [email protected] (Pradeep Kumar) Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
ficient of 50 W/m2K while right wall is also isothermal at same temperature as
of lower left wall and upper wall is adiabatic. All walls are opaque for radiation
except semitransparent window. The results reveal that the dynamics of both
the vortices inside the cavity change drastically with irradiation values and also
with the boundary conditions on the semitransparent window. The temperature
and Nusselt number increase multifold inside the cavity for combinedly conduc-
tively and radiatively adiabatic condition than the only conductively adiabatic
condition on the semitransparent window. Keywords: Semitransparent window; Natural convection; Collimated
beam; Symmetrical cooling; Irradiation; Bottom Heating;
1. Introduction
The knowledge of heat transfer in buoyancy driven flows is crucial in many
natural and engineering applications, like HVAC systems, electronic cooling. In
recent years, there has been increasing interest of the researchers in the analysis
5 of multi-physics problems involving all modes of heat transfer.
Many researchers [1, 2, 3, 4, 5] investigated the buoyancy driven flows in
2D enclosures heating from bottom and cooled either one side or both sides.
Acharya and Goldstein [6] studied the effect of internal energy sources on the
natural convection in an inclined square box. Osman et al. [7] investigated the
10 effect of non-newtonian fluid (i.e Bingham fluid) on natural convection, it was
observed that Nusselt number was smaller in Bingham fluids than Newtonian
fluid for the same Rayleigh and Prandtl numbers. A comprehensive review on
natural convection has been performed by Rahimi et al. [8] and Das et al. [9].
The interaction of radiation with natural convection has been investigated
15 by Mondal and Mishra [10], Liu et al. [11] and Kumar and Eswaran [12] in a 2-
dimensional cavity with different optical thicknesses and reported that there was
2 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
an considerable effect of radiation on the natural convection. The square cavity
has been selected in first two works whereas slanted cavity was used by Kumar
and Eswaran. Lari et al. [13] also showed that the radiation has considerable
20 effects on the natural convection even at low operating temperatures. Further
studies on the effect of different geometrical shapes, like square, triangle and
circle at the centre of the square cavity on the combined radiation and natural
convection were performed by Mezrhab et al. [14], Sun et al. [15] and Mukul
et al. [16], whereas, the study with circle inside the circular cavity has been
25 performed by Xu et al. [17].
The discreate ordinate (DO) method [18] was most acceptable and popular
method to solver radiative heat transfer equation before finite volume discreate
ordinate method (fvDOM) [19, 20] and now the researchers mostly use fvDOM
in the radiation field. The performance analysis of various methods for radiation
30 transfer equation (RTE) like FVM, DOM, P1, SP3 and P3 have studied by Sun
et al. [21], and reported that the P1 method consumed minimal time than
the other methods, however, it suffer in accuracy at low optical thickness of
medium whereas FVM gave better accuracy compared to DOM but took more
computation time. Assessment of natural convection with radiation for different
5 35 aspect ratios ranging from 1 to 6, and range of Rayleigh numbers 1.60 × 10 to
4.67 × 107 in rectangular enclosure have been studied numerically in [22] and
heat transfer correlations has been formulated. It has been reported that the
mean Nusselt number increased to 73.35 from 23.63 for aspect ratio 1 to 6.
Webb and Viskanta [23] performed an experiment to study the natural con-
40 vection with water as working fluid in an rectangular enclosure irradiated with
collimated beam from a side and kept opposite wall on constant temperature,
whereas other walls were kept adiabatic. The results showed that there was
formation of hydrodynamic boundary layers at the vertical walls. The authors
3 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
have also analysed numerically the same problem for the prediction of internal
45 heating in the fluid.
Anand and Mishra [24] investigated the radiative transfer equation for vari-
able refractive index for participating media. Ben and Dez [25] provided an
exact expression for the radiative flux for the emitting and absorbing semi-
transparent medium which has a linear refractive index variation. The above
50 researchers mostly considered the combined diffuse radiation with natural con-
vection, however, little work is available on collimated radiation. Apparently to
the present authors knowledge there is no much work focusing the interaction
of collimated beam radiation with natural convection.
Also, the walls of the geometries in all above works have been considered
55 as opaque which does not allow any energy enter into domain through radi-
ation mode of heat transfer. Whereas, semitransparent wall allows energy to
enter into the system by radiative mode of heat transfer but may or may not
allow to leave the system. This arises two heat flux boundary conditions on the
semitransparent window and such conditions are mostly encounter in practical
60 scenarios. In the present work two heat flux boundary conditions on the semi-
transparent wall for natural convection problem including radiation have been
investigated. This paper is outlined as follows; section 2 describes the prob-
lem statement followed by mathematical modelling and numerical schemes in
section 3. The validation and grid independent test are explained in section 4
65 and 5, respectively. Section 6 elaborates the results and discussion.Finally, the
conclusions of the present numerical study are provided in section 7.
2. Problem statement
Figure 1 depicts the geometry for present study. The ecludian co-ordinate
axis are along horizontal and vertical walls of cavity and origin is at lower left
4 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
Figure 1: Schematic diagram of cavity for the present problem with collimated beam incident at an angle of 450
70 corner of the cavity. The gravity force acts in negative Y direction. The left
wall of the cavity has been divided into two upper and lower parts in the ratio
of 4:6, where the lower part of the cavity is opaque while upper part is semi-
transparent (for the radiation energy) and the rest walls are considered to be
opaque. The thermal conditions on the lower left and right vertical walls are
75 constant temperature at 296 K while upper wall is thermally adiabatic. The
cavity is heated from bottom by convective heat transfer with heat transfer co-
efficient of 50 W/m2K and free stream temperature is 305 K. The combined
radiation and natural convection are considered inside the cavity. The working
fluid of the cavity is considered to be transparent for radiation energy and flow
5 80 is governed buoyancy force corresponds to Rayleigh number 10 and Prandtl
number (Pr=0.71). A collimated beam enters through upper left wall inside the
cavity at an angle of 450. This semitransparent wall may or may not allow the
radiation energy to leave the cavity assuming being low conductivity of semi-
transparent material, the energy does not leave the system through conduction
85 mode of heat transfer. Based on these two scenarios two thermal conditions on
the semitransparent wall are considered
case (A). Conductively Adiabatic: The semitransparent wall allows to leave
5 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
energy through radiation mode but not by conduction mode, i.e qc = 0, case (B) Combinedly conductively and radiatively adiabatic wall: The semi-
90 transparent wall does not allow the energy leave neither by conduction nor by
radiation heat transfer, i.e qc + qr = 0. The effect of above two boundary conditions on the fluid flow and heat
transfer characteristics inside the cavity has been studied for irradiation values
of 0, 100, 500 and 1000 W/m2.
95 3. Mathematical formulation and Numerical procedures
3.1. Mathematical formulation
The following assumptions have been considered for the mathematical mod-
elling of the problem;
1. Flow is steady, laminar, incompressible and two dimensional.
100 2. Flow is driven by buoyancy force that is modeled by Boussinesq approxi-
mation.
3. The thermophysical properties of fluid are constant.
4. The fluid may or may not participate in radiative heat transfer.
5. The refractive index of the medium is constant and equal to one.
105 6. The fluid absorbs and emits but does not scatter the radiation energy.
7. The transmissivity of semitransparent window is one and zero for other
walls.
Based on the above assumptions the governing equations in the Cartesian co-
ordinate system are given by
∂u i = 0 (1) ∂xi
6 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
2 ∂uiuj 1 ∂p ∂ ui = − + ν + gβT (T − Tc)δi2 (2) ∂xj ρ ∂xi ∂xj∂xj
∂u T k ∂2T 1 ∂q j = − R (3) ∂xj ρCp ∂xj∂xj ρCp ∂xi
∂qRi 110 where is the divergence of the radiative flux, which can be calculated as ∂xi
∂qRi = κa(4πIb − G) (4) ∂xi
where κa is the absorption coefficient, Ib is the black body intensity and G is the irradiation which is evaluated by integrating the radiative intensity (I) in
all directions, i.e.,
Z G = IdΩ (5) 4π
The intensity field inside the cavity can be obtained by solving the following
115 radiative transfer equation (RTE)
∂I(ˆr,ˆs) = κ I (ˆs) − (κ )I(ˆr,ˆs) (6) ∂s a b a
Where ˆr,ˆs is position and direction vectors, whereas s is path length. The
Navier-Stokes equation and temperature equations are subjected to boundary
conditions
Flow boundary condition
120 Cavity walls: ui=0
Thermal boundary conditions
1. Left wall (a) Lower part: Isothermal
T=296 K and
7 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
(a) (b)
Figure 2: Pictorial representation of (a) cell arrangement for finite volume method for partial differential equation and (b) Angular discretization for the radiative transfer equation
(b) Upper part: either qc = 0 or
125 qc + qr = 0
2. Right wall at x=1; T=296K
3. Bottom wall y=0 ; qconv = hfree(Tfree − Tw)
4. Top wall at y=1; qc + qr = 0
∂T R where qc = −k ∂n and qr = 4π I(rw)(ˆn · ˆs)dΩ
130 The radiative transfer equation (6) is subjected to following boundary con-
dition, all cavity wall (assumed black wall) except semitransparent window
Z 1 − w I(rw,ˆs) = wIb(rw) + I(rw,ˆs)|ˆn · ˆs|dΩ. π ˆn·ˆs>0 for ˆn · ˆs < 0 (7)
wheren ˆ is the surface normal and the emissivity of all walls is considered to
be 1.
The semitransparent window is subjected to collimated irradiation (Gco) of
2 135 value 1000 W/m . The boundary condition for RTE on semitransparent window
8 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
(b) (a)
Figure 3: Pictorial representation of (a) Diffuse reflection of an incident ray and diffuse emis- sion due to wall temperature (b) Diffuse emission and collimated transmission from a semi- transparent wall is
Z 0 1 − w I(rw,ˆs) = Ico(rw,ˆs)δ(θ − 45 ) + wIb(rw) + I(rw,ˆs)|ˆn · ˆs|dΩ. π ˆn·ˆs>0 for ˆn · ˆs < 0 (8) where δ(θ − 450) is Dirac-delta function,
0 1, if θ = 45 , δ(θ − 450) = (9) 0 0, if θ 6= 45 .
Ico is intensity of collimated irradiation and calculated from the irradiation value as below G I = co (10) co dΩ
Where dΩ is the collimated beam width. In the current work, the solid angle of discretized angular space (Fig. 2b) in collimated direction is considered as beam
9 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
width of the collimated beam. The pictorial representation of diffuse emission
and reflection and collimated beam radiation from the wall is shown in Fig. 3.
The collimated feature has been developed in OpenFOAM framework an open
source software and coupled with other fluid and heat transfer libraries. The
combined application have been used for numerical simulation. The OpenFOAM
uses the finite volume method (FVM) to solve the Navier-Stokes and the energy
equations. The FVM integrates an equation over a control volume (Fig. 2a) to
convert the partial differential equation into a set of algebraic equations in the
form X apφp = anbφnb + S (11) nb
where φp is any scalar and ap is central coefficient, anb coefficients of neighbour- ing cells and S is the source values, whereas, RTE eq (6) is converted into a
set of algebraic equations by double integration over a control volume and over
a control angle. The set of algebraic equations are solved by Preconditioned
bi-conjugate gradient (PBiCG) and the details of the algorithm can be found in
the book by Patankar [26] and Moukalled [27]. In the present simulation, linear
upwind scheme which is second order accurate has been used to interpolate face
centred value. The linear upwind scheme is given mathematically as
φp + ∇φ · ∇r, if fφ > 0, φf = (12) φnb + ∇φ · ∇r, if fφ < 0.
and fφ is the flux of the scalar φ on a face (Fig 2a).
3.2. Non-dimensional Parameters
The OpenFOAM simulation produces the results in dimensional quantities.
140 To explain the results in more general form, the simulated results are converted
into non-dimensional parameters. The scales for length, velocity, temperature,
10 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
and conductive and radiative fluxes are L, uo, (Tfree-Tc), κ(Tfree-Tc)/L and
4 p σTfree respectively, where uo = Lgβ(Tfree − Tc) is convective velocity scale. The non-dimensional quantities and parameters involved in the present prob-
145 lem are as follows,
u v x y T − T U = ,V = ,X = ,Y = , θ = c (13) uo uo L L Tfree − Tc
gβ(T − T )L3 ν Ra = free c , P r = (14) να α
The optical thickness is defined as τ = κaL and non-dimensional irradiation is given as
G G = 4 (15) σTfree
The NuC and NuR are conductive and radiative Nusselt numbers respectively and defined as
qCwL qRwL NuC = , NuR = (16) k(Tfree − Tc) k(Tfree − Tc)
150 Thus, the total Nusselt number is defined as below,
Nu = NuC + NuR (17)
4. Validation
In the absence of any standard benchmark test case for the present problem,
the validation has been performed in three steps, first, the standalone feature
of collimated beam irradiation problem, in second step, pure natural convec-
11 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
(b) (a)
Figure 4: Schematic for the (a) Test case for the collimated beam radiation and (b) Contour representing the collimated beam travelling normal to the wall
Figure 5: Validation results for pure convection and combined diffuse radiation with natural convection
155 tion problem which is heated from the bottom and in the third step combined
convection and radiation in defferentially heated cavity have been verified. The
collimated irradiation feature [28] has been tested in a square cavity as shown in
the Fig. (4a). The left side of the wall has s small window of non-dimensional
size 0.05 at a non-dimensional height of 0.6. The walls of square cavity are
160 black and cold and also medium is non-participating. A collimated beam is
12 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
irradiated on the window in 450 direction. It is expected that the beam would
travel in oblique direction of 450 angle without any attenuation and hit exactly
non-dimensional distance of 0.6 from left wall. Figure (4b) shows the contour
of irradiation which clearly shows the travel of collimated without any atten-
165 uation. For second step, fluid flow with heat transfer (without any radiation)
is validated against Aswatha et al. [29] and combined diffuse radiation and
natural convection in a cavity whose top and bottom walls are adiabatic and
vertical walls are isothermal at differential temperatures and radiatively opaque
has been validated against Lari et al. [13]. The present results for both the
170 cases (see Fig. 5) are in good agreement with the published results.
5. Grid independent test
Numerical solutions of Navier-Stokes and energy equation and radiation
transfer equation are sensitive to the spatial discretization. Additionally, radia-
tive transfer equation also requires angular space discretization which provides
175 directions along which radiation transfer equation is being solved. Thus, opti-
mum number of grids and directions have been obtained through independent
test study in two steps,
1. Spatial grids independence test: Three spatial grid sizes,i.e, 60×60, 80×80
and 100×100 are chosen to calculate the average Nusselt number on the
180 bottom wall for the present problem of natural convection. The Nusselt
number values calculated for above three grids arrangement on the bottom
wall are 6.544 6.634 and 6.659, respectively. The percentage error between
the first and second is 1.35%, whereas between second and third is 0.37%.
Thus, the spatial grid points i.e 80×80 is selected for further study.
185 2. Angular direction independence test: The polar discretization has no ef-
fect on the two-dimensional cases, thus OpenFOAM fixes the number of
13 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
Figure 6: Contours of collimated irradiation on semitransparent window at angle of 450
polar directions to 2, in one hemisphere of angular space, while azimuthal
considered directions are 3, 5, and 7 for the angular direction independent
were studied. The Nusselt numbers on 80×80 spatial grid and 2×3, 2×5,
190 2×7 are 6.946, 7.0 and 7.012, respectively. The percentage difference in
the first and second angular discretization is 0.7%, whereas in second and
third angular discreitization is 0.17%. Thus, finally nθ × nφ = 2 × 5 in one hemisphere angular space is selected.
6. Results and Discussion
195 Two scenarios of boundary conditions on semitransparent wall that are dis-
cussed in the section 2, have been simulated with collimated beam feature devel-
oped in OpenFOAM [30] framework with natural convection and comprehensive
results have been presented in this section.
6.1. Irradiation contour
0 200 A collimated beam in the direction of 45 is applied on the semitransparent
window and allowed to enter into the domain. The collimated beam in the
14 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
cavity has been depicted by irradiation contours in Fig. 6. Being transparent
medium inside the cavity, the collimated beam strikes on the bottom wall and
irradiation value remains constant through out the travel of the beam. The
205 variations of irradiation value are seen at the edge of the irradiation contours
that is basically due to rendering of the contours otherwise, contour change in
the travel zone and outside should be like heavy side step function. The beam
strikes at the non-dimensional distance 0.6 from the left corner and strike length
of the collimated beam is 0.4 on the bottom wall which are same as window non-
210 dimensional height and width. The effect of different values of irradiation and
boundary conditions on semitransparent wall on the fluid flow and heat transfer
characteristics will be explained in the subsequent sections.
6.2. Case A: Conductively Adiabatic Condition
The semitransparent wall behaves as only conductively adiabatic and allows
215 the energy leaves by radiation mode. The detailed analysis of fluid flow and
heat transfer are presented in the following section.
6.2.1. Fluid Flow Characteristics
The present problem consists two symmetrical counter rotating vortices in-
side the cavity when there would not have been collimated beam and the two
220 vertical walls are isothermal and non-uniform heating at bottom [31]. However,
asymmetricity has been introduced by two ways (1) semitransparent vertical
wall have been made conductively adiabatic and (2) there is collimated irradia-
tion on the semitransparent window. The asymmetric is high where size of left
vortex is very small compared to right side vortex without any irradiation as
225 shown in Fig. 7a. This asymmeticity decreases with increase of irradiation val-
ues of 100, 500 as depicted in 7b, 7c 7d, respectively. The left vortex increases
with the increase of irradiation value and the size of right vortex decreases in the
15 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
(a) (b)
(c) (d)
Figure 7: Contours of non-dimensional stream function for collimated irradiation on the semi- transparent window for the values of (a)G=0 (b)G=100 (c)G=500 and (d)G=1000 W/m2
similar proportion. This could be owing to the fact that being medium trans-
parent, whole irradiation strikes on the bottom wall, this causes increase of the
230 temperature of the bottom wall, thus, increase in the buoyancy force. The net
of buoyancy and momentum force is now more in upward direction, therefore,
the size of right vortex decreases. This net force is more in upward direction
with the increase of the irradiation value. The maximum non-dimensional val-
ues of stream function for both the vortices for various values of irradiation are
235 depicted in Table 1. The maximum non-dimensional value of stream function
for right vortex remains almost constant while it increases for the left vortex.
The flow rate in the left vortex increases with increase in the value of stream
function and size of the left vortex while it decreases in the right vortex.
16 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
Figure 8: Variation of non-dimensional vertical velocity along the horizontal lines at the mid- height of the cavity for various values of irradiation
Figure 8 depicts the variation of non-dimensional vertical velocity in the hor-
240 izontal direction at the mid height of the cavity for various values of irradiation.
The direction of vertical velocity is downward on both the vertical walls clearly
indicate that hot fluid descends from these isothermal wall. Furthermore, the
values of maximum vertical velocity in downward direction is lower near to left
wall compared to right wall while maximum non-dimensional vertical velocity
245 in downward direction increases with increase of the irradiation value. This
increment is more near to left wall than the right wall, also the location of
maximum downward vertical velocity remain same from both the vertical walls,
i.e., around non-dimensional distance of 0.15 from both the vertical walls. The
maximum non-dimensional vertical velocity values in downward direction near
250 to left side wall are 0.08, 0.09, 0.2 and 0.22 for irradiation values of 0, 100, 500,
1000 W/m2, respectively, while same values near to the right side walls are 0.3,
Table 1: Non-dimensional maximum stream function values for the various values of collimated irradiation
Irradiation(G) 0 100 500 1000 Right vortex -0.069 -0.069 -0.067 -0.069 Left vortex 0.012 0.015 0.03 0.036
17 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
(a) (b)
(c) (d)
Figure 9: Contours of non-dimensional temperature for collimated irradiation on the semi- transparent window of values (a)G=0 (b)G=100 (c)G=500 and (d)G=1000 W/m2
0.3, 0.32 and 0.38, respectively. There is also gain in maximum value of vertical
velocity in upward direction, but unlike to the downward vertical velocity, the
location of maximum upward velocity are different for different values of irradi-
255 ation. These locations are at non-dimensional distance of 0.21, 0.21, 0.405, and
0.44, respectively from the left wall and maximum values of non-dimensional
vertical maximum velocity 0.3, 0.3, 0.35 and 0.42 for irradiation values of 0,
100, 500, 1000 W/m2, respectively.
6.2.2. Heat transfer characteristics
260 Being stream lines asymmetric (Fig. 7), the isothermal lines are also asym-
metric (Fig. 9). The isothermal lines at core are bent toward the left wall due
18 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
to fact that the left vortex is smaller than the right vortex. Furthermore, the
isotherms lines are clustered near to the bottom and isothermal walls whereas
almost uniform temperature is spread in the core of the cavity near to adiabatic
265 top wall. The isotherm lines are perpendicular to the semitransparent wall
reveals that the wall is conductively adiabatic whereas the isotherms are not
perpendicular to the upper wall, because upper wall is combinedly conductively
and radiatively adiabatic. Table 2 represents the maximum non-dimensional
temperature inside cavity and this increases with increase of irradiation value.
270 This maximum non-dimensional temperature exists at the junction point of two
vortices that creates a hot spot in the area of irradiation values of 0 and 100
W/m2 whereas this maximum non-dimensional temperature exist at the strike
length of collimated beam on the bottom wall for case of irradiation values of
500 and 1000 W/m2 cases. One interesting point to notice is that isotherm
275 lines are more clustered at the strike length of the collimated beam and also the
non-dimensional temperature increases beyond to 1 for irradiation values of 500
to 1000 W/m2.
The variation of non-dimensional temperature on the various walls and also
along the horizontal direction at the mid height of the cavity have been shown in
280 Fig 10. The non-dimensional temperature at the bottom wall increases from the
left corner and reaches to peak value of 0.82 then decreases upto the right corner
for case of G = 0 while there is small rise in temperature at the strike length
of collimated beam for G=100 W/m2. Whereas there is drastic rise in non-
dimensional temperature at the strike length of collimated beam for irradiation
2 2 285 value of 500 W/m and 1000 W/m and maximum non-dimensional temperature
Table 2: Non-dimensional maximum isothermal values for the various values of collimated irradiation
Irradiation(G) 0 100 500 1000 Maximum 0.820 0.830 1.095 1.426
19 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
(a) (b)
(c) (d)
Figure 10: Variation of non-dimensional temperature at (a) bottom wall (b) horizontal line at mid-height of cavity (c) left wall and (d) top wall for various values of collimated irradiation
reaches to 1.2 and 1.47, respectively (see Fig. 10a) at the bottom wall. The
temperature rise is not that much drastic in the path of collimated beam at
the mid height of the cavity (Fig. 10b). The global maxima in the temperature
curve is near to left wall at mid-height of the cavity that shifts towards right with
290 the increase of the irradiation and the maximum non-dimensional temperatures
are at 0.4, 0.42, 5.2 and 5.6 at non-dimensional distance from left is 0.2, 0.2,
0.38, 0.42 for irradiation values of 0, 100, 500 and 1000 W/m2, respectively. The
non-dimensional temperature remain zero on the left wall upto the height of 0.6
because of isothermal condition, afterwards the non-dimensional temperature
295 increases all of sudden (Fig. 10c). The maximum non-dimensional temperature
rise are 0.38, 0.39, 0.41 and 0.5 for irradiation values of 0, 100 , 500 and 1000
20 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
W/m2, respectively and remain more or less constant afterwards on the left
semitransparent wall. The non-dimensional temperature remains constant and
matches with temperature rise on semitransparent vertical wall for most of the
300 length on upper wall, only decrease towards end to match the temperature of
the isothermal right wall (see Fig. 10d)
6.2.3. Nusselt number
The variations of conduction, radiation and total Nusselt numbers on the
bottom wall are shown in Fig. 11(a), (b) and (c), respectively, for various values
305 of irradiation. The conduction Nusselt number is very high at the end of the
walls and monotonically decreases from both the ends and reached to minimum
(a) (b)
(c)
Figure 11: Variation of (a) conduction (b) radiation and (c) total Nusselt number on the bottom wall for various values of collimated irradiation on semitransparent window
21 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
values at different locations on the wall for different values of irradiation; its
location is at the junction of two vortices for irradiation value of 0 and 100
W/m2 and before the strike point from left for irradiation values of 500 and 1000
2 310 W/m afterward there is sudden rise in the value of conduction Nusselt numbers.
The minimum conduction Nusselt numbers are same for the irradiation values
of 0 and 100 W/m2 while for irradiation values of 500 and 1000 W/m2 its
value are 2 and 0 respectively. While maximum conduction Nusselt numbers
are also same for irradiation value of 0 and 100 W/m2 which is 15 while its
2 315 value is 20 and 24 for irradiation value of 500 and 1000 W/m , respectively, this
maximum Nusselt number is found near to right corner of the wall. The radiative
Nusselt number is almost zero upto the strike point from left corner of wall and
sudden increase in the radiation Nusselt number has been observed over the
strike length of the collimated beam on the bottom wall. The negative radiative
320 Nusselt number indicates that energy is going out of the domain by radiation
mode of heat transfer. The increase of radiative Nusselt number happens with
increase in irradiation value. Also, the radiative Nusselt number is zero over
whole curve length of the bottom wall for irradiation value 0 W/m2 indicates the
contribution of the diffuse radiation is almost negligible in the present problem.
325 Thus, the total Nusselt number is governed by conduction Nusselt number upto
the non strike length of the collimated afterword, it is governed by radiation
over the strike length of beam, if the irradiation values is sufficiently high,
here this happens for G > 500 W/m2 (Fig. 11c). Similarly, the conduction,
radiation and total Nusselt number on the left wall are shown in Fig. 12(a),
330 (b) and (c), respectively. The conduction Nusselt number decreases drastically
upto non-dimensional height of 0.1 afterward its starts increasing slowly for
irradiation values of 0 and 100 W/m2 while fast increment happens for higher
irradiation values of 500 and 1000 W/m2 upto the lower isothermal wall height,
22 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
afterwards, sudden decrease to zero due to conductively adiabatic wall condition
335 on the upper semitransparent wall. The negative conduction Nusselt number
indicate that energy goes out through conduction from lower isothermal wall.
The radiative Nusselt number is almost zero on the isothermal wall further
emphasises that contribution of diffuse radiation is almost zero. While there
is sudden increase of radiative Nusselt number on the upper semitransparent
340 wall and remain constant over the height due to collimated irradiation. The
positive radiative Nusselt number indicates that energy enters into the domain
through radiative mode of heat transfer. The total Nusselt number is linear
sum of both conduction and radiation Nusselt numbers, thus, conduction is
dominant on lower isothermal wall and radiation on the upper semitransparent
345 wall. Thus, net energy leaves out through isothermal wall and comes into the
domain through upper semitransparent wall.
Further the conduction, radiation and total Nusselt number on right side
wall are shown in Fig 13(a), (b) and (c), respectively. The conduction Nus-
selt number decreases sharply to a value of 2 to a height of 0.1 afterwards, it
350 almost remain constant throughout the height of wall for irradiation value of
100 W/m2 and small increment is found for irradiation value of 500 and 1000
W/m2. Whereas, the radiation Nusselt number remains almost constant for the
irradiation values of 0 and 100 throughout the wall but large value of Nusselt
number is found at the lower height of the wall then the value decreases to 2,
355 within a non-dimensional height of 0.1 for the irradiation values of 500 and 1000
W/m2, afterthat, the value almost remains constant and the Nusselt number
value is very small. The total Nusselt number of the right wall is the linear com-
bination of two Nusselt numbers i.e conduction and radiation Nusselt number.
It can be clearly understood that variation of total Nusselt number is similar to
360 conduction Nusselt number over whole height of right wall. The negative values
23 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
(a) (b)
(c)
Figure 12: Variation of (a) conduction (b) radiation and (c) total Nusselt number on the left wall for various values of collimated irradiation on semitransparent window
show the energy leaves by both conduction and radiation mode of heat transfer
from the right wall.
The average Nusselt number on the various walls of the cavity for different
values of irradiation is shown in Table 3. The total average Nusselt number
365 decreases with the increase of irradiation value on bottom wall while average
conduction Nusselt number increases and radiation Nusselt number changes its
sign from being positive on low irradiation value to negative for higher irra-
diation value. The left wall has two portions, lower part is isothermal where
average of both conduction and radiation Nusselt numbers remain almost con-
370 stant and negative, while upper portion being semitransparent, the conduction
Nusselt number is zero due to conductively adiabatic condition and radiative
24 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
(a) (b)
(c)
Figure 13: Variation of (a) conduction (b) radiation and (c) total Nusselt number on the right wall for various values of collimated irradiation on semitransparent window
Nusselt number increases with increase of irradiation value. Furthermore total
average Nusselt number first being negative then becomes positive with increase
of irradiation value on the left wall. The average conduction and radiation Nus-
375 selt numbers on the right wall increases with increase of irradiation values, thus
the total Nusselt number also increases with negative sign.
25 Total
Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar Radiation Right wall Conduction Total Radiation Semitransparent wall Conduction Left wall Radiation Isothermal wall Conduction Total Radiation Bottom wall Conduction Table 3: Average Nusselt number on different walls for various values of irradiation for conductively adiabatic semitransparent wall (G) Irradiation 01005001000 5.403 5.678 6.792 8.15 0.923 0.318 -3.435 -7.98 6.326 5.996 3.357 -2.138 -2.177 -2.702 0.17 -2.852 -0.435 -0.444 -0.478 0 0 -0.521 0 0 0.026 1.057 5.188 -2.547 10.326 -1.564 2.008 -2.975 -3.531 -3.991 6.953 -5.244 -0.791 -0.901 -1.335 -3.766 -4.432 -1.879 -5.326 -7.123
26 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
(a) (b)
(c) (d)
Figure 14: Contours of non-dimensional stream function for collimated irradiation of (a)G=0 (b)G=100 (c)G=500 and (d)G=1000 W/m2 on semitransparent window
6.3. Case B: Combinedly conductively and radiatively adiabatic condition
In this section, the fluid flow and heat transfer characteristics have been
extensively studied when combinedly conductively and radiatively adiabatic
380 boundary condition has been applied on the semitransparent window. In this
condition even energy does not leave the system by radiaitve mode of heat
transfer through semitransparent window.
6.3.1. Fluid flow characteristics
The contours of non-dimensional stream function for various values of irra-
385 diation on the semitransparent window have been presented in Fig. 14. The
stream function contours for zero irradiation (Fig.14a) is similar to case A (Fig.
27 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
Table 4: Non-dimensional maximum stream function values for the various values of collimated irradiation on semitransparent window
Irradiation(G) 0 100 500 1000 Right -0.069 -0.057 -0.047 -0.047 Left 0.011 0.021 0.037 0.039
7a) because of negligible diffuse radiation is present in current problem. How-
ever, the behaviour of two vortices changes drastically with the collimated irradi-
ation on the semitransparent window. The left side vortex always remains upto
390 the height of isothermal wall for all values of irradiation, however, its width
increases with the increase of irradiation values. Being combinedly adiabatic
semitransparent wall, it is expected that it will be hotter, thus fluid rises from
semitransparent wall and only lose energy to the right side of cold isothermal
wall. Therefore, area occupied by right side vortex becomes quite large includ-
395 ing whole area above to the lower isothermal wall and right side vortex area.
At lower irradiation value (G = 100W/m2) the orientation of right side vortex
is diagonal, whereas the orientation of right side vortex becomes straight with
increase of irradiation value and layered flow happen on the upper part (above
to the lower left wall) of the cavity however, this layered flow is connected to
400 the right side vortex. Table 4 shows the maximum value of non-dimensional
stream function for these two vortices for various values of irradiation. The
non-dimensional stream function value keeps on increasing for left vortex with
increase of irradiation value while its decreases for right vortex with increase of
irradiation value upto 500 W/m2 then it remains constant. Furthermore, the
405 area occupied by left vortex also increases which indicates that the volume flow
rate increases in left vortex with increase of irradiation value.
The variation of non-dimensional vertical velocity in the horizontal direction
at the mid height of the cavity is depicted in Fig. 15. The interesting fact to
notice is that the maximum vertical velocity in downward direction increases
28 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
Figure 15: Variation of non-dimensional vertical velocity along the horizontal lines at the mid-height of the cavity for various values of irradiation on semitransparent window
2 410 for the left vortex upto irradiation values of 500 W/m then slight decrements
has been noticed for irradiation value 1000 W/m2 while the maximum vertical
velocity in downward direction remains almost constant for the right vortex
for all values of irradiation. Furthermore, maximum vertical velocity in upward
direction almost remains constant for all values of irradiation, while, its location
415 keeps on shifting to right with increase in the value of irradiation. The value of
this maximum non-dimensional vertical velocity in upward direction is 0.3 while
maximum non-dimensional vertical velocity in downward direction is found right
vortex and its values is 0.3.
6.3.2. Heat transfer characteristics
420 The contours of non-dimensional temperature for various values of irradia-
tion are shown in Fig. 16. Although, the stream function contours show no
difference between case A as B, (see Fig. 7a and 14a) for irradiation value zero,
the non-dimensional temperature contours for case B shows the difference at the
upper part of the cavity near to semitransparent window, else it remain similar
425 to case A. The clustering of isothermal lines are near to isothermal walls and
dense clustered lines are visible near to bottom wall for irradiation value of 0
29 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
(a) (b)
(c) (d)
Figure 16: Contours of non-dimensional isothermal for collimated irradiation of (a)G=0 (b)G=100 (c)G=500 and (d)G=1000 W/m2 on semitransparent window
and 100 W/m2. However, scenario changes drastically with the little increase in
collimated irradiation. Now, the clustering of isothermal lines happens on semi-
transparent window and upper adiabatic wall, also, the clustering on isothermal
430 and bottom walls have decreased. This phenomenon increases with increase of
irradiation values. For high value of irradiation, the non-dimensional tempera-
ture is more uniform on lower part of the cavity than the upper part of cavity
where high temperature exists inside the cavity unlike to case A. Table 5 shows
the maximum non-dimensional temperature inside the cavity. The maximum
435 non-dimensional temperature increases to 5.758 for irradiation value of 1000
W/m2 which exist at the junction point of semitransparent and above adiabatic
walls, unlike to case A which always occur on the bottom wall.
30 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
The variation of non-dimensional temperature on the bottom wall, in hor-
izontal direction at mid-height of the cavity, left and top walls are shown in
440 Fig. 17 (a), (b), (c) and (d), respectively. The temperature increases slowly on
the bottom wall from both the corners for lower values of irradiation. While
spatial rate of increase of temperature is higher on the right corner of the wall
for higher value of irradiation and sudden decrease in non-dimensional temper-
ature is found just after the collimated beam strike zone. There is hardly any
445 difference in temperature found near to left corner of the wall with irradiation
values. Unlike to the temperature variation on the bottom wall, there is tem-
perature difference in the horizontal direction for various values of irradiation
(a) (b)
(c) (d)
Figure 17: Variation of non-dimensional temperature at (a) bottom wall (b) horizontal line at mid-height of cavity (c) left wall and (d) top wall for various values of collimated irradiation on semitransparent window
31 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
at mid height of the cavity. The global maxima in the curve exists at the left for
lower values of irradiation, whereas, there are three local maxima are found for
450 higher values of irradiation and now, global maxima of temperature curve exists
on the right. The non-dimensional temperature remains constant upto height of
isothermal portion of the left wall, whereas, sudden increase in temperature has
been noticed on the upper semitransparent window. The spatial rate of increase
of temperature increases with increase of irradiation value. The continuity in
455 the temperature curve on junction of left and top wall is also observed (Fig.
17 (c) and (d), afterwards temperature decreases along the horizontal direction
on the top wall and matches temperature on right wall (isothermal wall). The
spatial rate of decrease of temperature decreases with decrease in irradiation
value.
460 6.3.3. Nusselt number characteristics
The conduction, radiation and total Nusselt number variation on the bottom
wall are presented in the Fig. 18 (a), (b) and (c), respectively for various values
of irradiation. The conduction Nusselt number is almost same on both the
ends of the bottom wall for the irradiation values of 0 and 100 W/m2, whereas
465 this trend changes with the increase in the irradiation value. The conduction
Nusselt number remains almost same on the left corner of the bottom wall, and
then slowly decreases to the minimum value at a non-dimensional distance of
0.3 from the left side wall for irradiation values 0 and 100 W/m2, whereas this
is minimum at a non-dimensional distance of 0.55 from the left corner for the
2 470 irradiation values of 500 and 1000 W/m , and then there is sudden increase in
the Nusselt number that reaches to the maximum value of 21 and 27 on right
corner of the bottom wall for the irradiation values of 500 and 1000 W/m2,
respectively. The radiation Nusselt number is almost zero over the entire wall
for the irradiation value 0 W/m2, whereas, this trend is almost followed till the
32 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
(a) (b)
(c)
Figure 18: Variation of (a) conduction (b) radiation and (c) total Nusselt number on the bottom wall for various values of collimated irradiation on semitransparent window
475 strike point, then there is increment observed in the radiation Nusselt number
for any non-zero value of irradiation. This increment is more pronounce for the
irradiation values of 500 and 1000 W/m2, where the most amount of energy
leaves by radiation mode of heat transfer from the bottom wall. The total
Nusselt number graphs follows the conduction Nusselt number graph till the
480 strike point and then the radiation dominates over strike length of the beam.
The variation of conduction, radiation and total Nusselt numbers on the ver-
tical left wall which also contains the semitransparent wall are shown in Fig. 19.
The conduction Nusselt number decreases drastically with height and reaches to
minimum value 3 at non-dimensional height of 0.1; afterwards it remains almost
485 constant on the isothermal left wall for lower values of irradiation, however, it
33 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
starts increasing for higher value of irradiation. There is sudden peak in the
conduction Nusselt number near the junction point of isothermal wall and semi-
transparent wall where adiabatic condition is applied. Afterward, it decreases
to further minimum value 0 and remain constant through out the height of semi-
490 transparent wall for irradiation value of zero and little increase in value for 100
W/m2, whereas, similar peak with large value is observed in conduction Nusselt
number near to junction point. After the peak the conduction Nusselt number,
further increases slowly over the height of semitransparent wall and drastically
increases for irradiation values of 500 and 100 W/m2. The negative conductive
495 Nusselt number indicates that the heat transferred from the system through
conduction mode of heat transfer. The radiative Nusselt number (Fig. 19b) is
almost zero over the height of isothermal wall whereas it increases sudden over
height of the semitransparent wall and increase with the irradiation value on
semitransparent wall.
500 The total Nusselt number which is a linear sum of conduction and radi-
ation Nusselt numbers is mostly governed by conduction Nusselt number on
isothermal wall because of diffuse radiation. The total Nusselt number is zero
on the semitransparent wall because of combinedly conductive and radiative
adiabatic condition. The conduction Nusselt number is equal and opposite to
505 radiative Nusselt number over the semitransparent wall. It can be verified by
conduction and radiation Nusselt number (Fig. 19(a) and (b)) curves. Total
Nusselt number also has a peak near the junction point of the isothermal and
the semitransparent wall.
The conduction, radiation and total Nusselt number variations on the right
510 wall are depicted in Fig. 20(a), (b) and (c), respectively. The conduction Nus-
selt number decreases drastically upto non-dimensional height of 0.1 afterward
it remain almost constant for irradiation value 0 W/m2 while it increases slowly
34 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
for lower value of irradiation and faster for higher value of irradiation over the
rest height of the right wall. The radiative Nusselt number is very small for
2 515 irradiation values of 0 and 100 W/m while it has significant value for irradi-
ation value of 500 and 1000 W/m2. It shows that the heat transfer by diffuse
radiation increases with the increase of collimated irradiation. Both conductive
and radiative Nusselt numbers are negative which indicates that energy leaves
the cavity by both the modes of heat transfer. The total Nusselt number is
520 the linear sum of conduction and radiation Nusselt number, thus, conduction is
being dominated mode of heat transfer in the present scenario, the total Nusselt
number variation is mostly similar nature as of conduction Nusslet number only
its values little is higher than the conduction Nusselt number due to additional
(a) (b)
(c)
Figure 19: Variation of (a) conduction (b) radiation and (c) total Nusselt number on the left wall for various values of collimated irradiation on semitransparent window
35 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
(a) (b)
(c)
Figure 20: Variation of (a) conduction (b) radiation and (c) total Nusselt number on the right wall for various values of collimated irradiation on semitransparent window of radiative Nusselt number.
36 Total -3.784 -5.138 -10.173 -16.735
Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar -0.802 -1.208 -2.656 -4.384 Radiation Right wall -3.93 -2.982 -7.523 -12.351 Conduction -5.09 Total -2.782 -3.314 -6.908 0.06 1.386 6.378 12.495 Radiation -0.06 -1.386 -6.378 -12.495 Semitransparent wall Conduction Left wall -0.454 -0.537 -0.864 -1.267 Radiation Isothermal wall -2.328 -2.777 -4.226 -5.641 Conduction 6.595 5.788 2.596 Total -1.279 1.202 0.045 -4.451 -10.013 Radiation Bottom wall 5.393 5.743 7.047 8.734 Conduction 0 100 500 (G) 1000 Irradiation Table 5: Average Nusselton number semitransparent on wall different walls for various values of irradiation for combinedly conductively and radiatively adiabatic condition
37 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
525 Table 6 represents the average value of Nusselt number on different walls of
cavity for various values of irradiation. The average conduction Nusselt num-
ber increases with the irradiation value while average radiation Nusselt number
is positive for lower irradiation value then becomes negative making decreases
in total Nusslet number with irradiation values on the bottom wall. At high
530 irradiation value, the energy also starts leaving from the bottom wall. The left
wall consists of two portions (1) lower isothermal wall (2) upper combinedly
conductively and radiatively adiabatic wall. On lower isothermal wall both the
conductive and radiative Nusselt numbers increases with the increase of irradi-
ation value and similar on the upper semitransparent wall, but the conduction
535 Nusselt number is same and opposite to radiative Nusselt number makes upper
semitransparent wall combinedly conductively and radiatively adiabatic. The
conduction, radiation and total Nusselt numbers behaviour on right side wall
is similar to the lower part of the left side wall where the Nusselt numbers are
negative and increases with increase of irradiation.
540 7. Conclusions
Two thermal adiabatic boundary conditions on semitransparent window
have been investigated with diffuse/collimated irradiation on a natural convec-
tion in a cavity for various values of irradiation. These two thermal adiabatic
boundary conditions arise based on the fact that the whether semitransparent
545 window allows the energy leaves the cavity by radiation mode of heat transfer
or not assuming being low thermal conductivity of semitransparent material,
the energy does not leave by conduction mode of heat transfer. Thus, in this
way, the semitransparent window may behave conductively adiabatic (case A)
or combinedly conductively and radiatively adiabatic (case B) and the following
550 conclusion are drawn
38 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
1. The dynamics of two vortices change with the change in irradiation values.
There is increase in size and stream function values of the left vortex,
while size of the right vortex decreases and stream function values remain
constant with increase of irradiation value for case A, whereas planner
555 flow exists at top of the cavity and left vortex remains in the lower part
of the cavity for higher values of irradiation for case B. 2. Maximum downward and upward vertical velocities at mid height of the
cavity is higher in case A than case B. 3. The clustering of isothermal lines are near to the isothermal and bottom
560 walls for all values of irradiation for case A while this clustering happen
near to semitransparent window and upper adiabatic wall for higher values
of irradiation for case B. 4. The non-dimensional temperature rises multi-fold inside the cavity for case
B than to case A and this high rise in temperature exist near to junction
565 point of semitransparent window and upper adiabatic wall for case B while
at strike length of collimated beam on bottom wall for case A. 5. The diffuse radiation has less influence in the present problem. The Nus-
selt number is majorly dominated by conduction Nusselt number upto the
non-strike length of collimated irradiation, afterward it is dominated by
570 collimated beam radiation.. 6. The total Nusselt number decreases with increase of irradiation on the
bottom wall while it increases on both the vertical walls for both the
cases. The Nusselt number becomes negative on bottom wall for higher
values of irradiation for case B while it remains positive for case A.
575 Acknowledgements
The authors greatly acknowledge the financial support provided by Science
and Engineering Research Board (SERB) (Statutory Body of the Government
39 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
of India) via Grant. No: ECR/2015/000327 to carry out the present work.
Declaration of interest
580 The authors declare that they have no known financial interests or personal
relationships that could have appeared to influence the work reported in this
paper.
References
[1] K.E. Torrance, J.A. Rockett, Numerical study of natural convection in an
585 enclosure with localized heating from below-creeping flow to the onset of
laminar instability, J.Fluid Mech, part 1 36 (1969) 33–54. doi:10.1017/
S0022112069001492.
[2] B. Calcagani, F. Marsili, M. Paroncini, Natural convective heat transfer in
square enclosures heated from below, App. Thermal engineering 25 (2005)
590 2522–2531. doi:10.1016/j.applthermaleng.2004.11.032.
[3] M.M. Ganzorolli, L.F. Milanez, Natural convection in rectangular en-
closures heated from below and symmetrically cooled from the sides,
Int.J. Heat mass Transfer, (6) 36 (1995) 1063–1073. doi:10.1016/
0017-9310(94)00217-J.
595 [4] Orhan Aydin, Wen-Hei Yang, Natural convection in enclosures with lo-
calized heating from below and symmetrical cooling from sides, Int. J
of Num. Methods for Heat & Fluid flow, No.5 10 (2000) 518–529. doi:
10.1108/09615530010338196.
[5] M. Sathiyamoorthy, Tanmay Basak, S. Roy, I. Pop, Steady natural con-
600 vection flows in a square cavity with linearly heated side wall(s), Int.
40 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
J. of Heat and Mass Transfer 50 (2007) 766–775. doi:10.1016/j.
ijheatmasstransfer.2006.06.019.
[6] S. Acharya, R.J. Goldstein, Natural convection in an externally heated
vertical or inclined square box containing internal energy sources, J. Of
605 Heat Transfer 107 (1985) 855–866. doi:10.1115/1.3247514.
[7] Osman Turan, Robert J. Poole, Nilanjan Chakraborty, Boundary condition
effects on natural convection of bingham fluids in a square enclosure with
differentially heated, Computational Thermal Sciences 4(1) (2012) 77–97.
doi:10.1615/ComputThermalScien.2012004759.
610 [8] Alireza Rahimi, Ali Dehghan Saee, Abbas Kasaeipoor, DEmad, Hasani
Malekshah, A comprehensive review on natural convection flow and heat
transfer the most practical geometries for engineering applications, Int. J.
of Numerical Methods for Heat and Fluid Flow 29 Issue: 3 (2019) 834–877.
doi:10.1108/HFF-06-2018-0272.
615 [9] Debayan Das, Monisha Roy, Tanmay Basak, Studies on natural con-
vection within enclosures of various (non- square) shapes-a review, Int.
J. of Heat and Mass Transfer 106 (2017) 356–406. doi:10.1016/j.
ijheatmasstransfer.2016.08.034.
[10] B. Mondal, S. C. Mishra, Simulation of natural convection in the presence
620 of volumetric radiation using the lattice boltzmann method, Num.Heat
Transfer,part-A 55 (2009) 18–41. doi:10.1080/10407780802603121.
[11] L. H. Liu, H. C. Zhang, H. P. Tan, Monte carlo discrete curved ray-tracing
method for radiative transfer in an absorbing-emitting semi-transparent
slab with variable spatial refractive index, J.of Quantative Spectroscopy
625 and Radiative Transfer 84 (2004) 357–362. doi:10.1016/S0022-4073(03)
00186-9.
41 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
[12] P. Kumar, V. Eswaran, The effect of radiation on natural convection in
slanted cavities of angle 450 and 600, Int.J.Thermal Science 67 (2013) 96–
106. doi:10.1016/j.ijthermalsci.2012.12.009.
630 [13] K. Lari, M. Baneshi, S. G. Nassab, A. Komiya, S. Maruyama, Combined
heat transfer of radiation and natural convection in a square cavity con-
taining participating gases, Int.J.Heat Mass Transfer 54 (2011) 5087–5099.
doi:10.1016/j.ijheatmasstransfer.2011.07.026.
[14] A. Mezrhab, H.Bouali, H. Amaoui, M. Bouzidi, Compuations of combined
635 natural convection and radiation heat transfer in a cavity having a square
body at its centre, App.Energy 83 (2006) 1004–1023. doi:10.1016/j.
apenergy.2005.09.006.
[15] Hua Sun, Eric Chenier, Guy Lauriat, Effect of surface radiation on the
breakdown of steady natural convection flows in a square, air-filled cavity
640 containing a centred inner body, App. Thermal Engineering 31 (2011) 252–
1262. doi:10.1016/j.applthermaleng.2010.12.028.
[16] Mukul Paramanda, Salman Khan, Amaresh Dalal, Ganesh Natarajan,
Critical assessment of numerical alogorithms for convective-radiative heat
transfer in enclosures with different geometries, Int.J. of Heat and Mass
645 Transfer 108 (2017) 627–644. doi:10.1016/j.ijheatmasstransfer.
2016.12.033.
[17] Xu Xu, Gonggang Sun, Zitao Yu, Yacai Hu, Liwu Fan, Kefa Cen, Nu-
merical investigation of laminar natural convective heat transfer from
a horizontal triangular cylinder to its concentric cylindrical enclosure,
650 Int. J. Heat and Mass Transfer 52 (2009) 3176–3186. doi:10.1016/j.
ijheatmasstransfer.2009.01.026.
42 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
[18] John C. Chai, HaeOk S. Lee, Suhas V. Patankar, Ray effect and false
scattering in the discrete ordinates method, Numerical Heat Transfer, Part
B 24 (1993) 373–389. doi:10.1080/10407799308955899.
655 [19] G. D. Raithby, E. H. Chui, A finite-volume method for predicting a radiant
heat transfer in enclosures with participating media, Trans. of the ASME,
Journal of Heat Transfer 112 (1990) 415–423. doi:10.1115/1.2910394.
[20] E. H. Chui, G. D. Raithby, Computation of radiant heat transfer
on a nonorthogonal mesh using the finite-volume method, Numerical
660 Heat Transfer, PartB: Fundamental 23 (1993) 269–288. doi:10.1080/
10407799308914901.
[21] Yujia Sun, Xiaobing Zhang, J. R. Howell, Assessment of different radia-
tive transfer equation solvers for combined natural convection and radia-
tion heat transfer problems, J. of Quantitative Spectroscopy and Radiative
665 Transfer 194 (2017) 31–46. doi:10.1016/j.jqsrt.2017.03.022.
[22] Hakan Karatas, Taner Derbentli, Natural convection and radiation in rect-
angular cavities with one active vertical wall, Int. J of Thermal sciences
123 (2018) 129 – 139. doi:10.1016/j.ijthermalsci.2017.09.006.
[23] B. W. Webb, R. Viskanta, Radiation-induced buoyancy driven flow in rect-
670 angular enclosures: Experiment and analysis, J.of Heat Transfer 109 (1987)
427–433. doi:10.1115/1.3248099.
[24] N. Anand Krishna, S. C. Mishra, Discrete transfer method applied to ra-
diative transfer in variable refractive index, J.of Quantative Spectroscopy
and Radiative Transfer 102 (2006) 432–440. doi:10.1016/j.jqsrt.2006.
675 02.024.
43 Collimated beam on Semitransparent Wall G chanakya, Pradeep Kumar
[25] P. Ben Abdallah, V. Le Dez, Radiative flux field inside an absorbing-
emitting semi transparent slab with a variable refractive index at radiative
conductive coupling, J.of Quantative Spectroscopy and Radiative Transfer
67(2) (2000) 125–137. doi:10.1016/S0022-4073(99)00200-9.
680 [26] S. V. Patankar, Numerical heat transfer and fluid flow, Hemisphere Pub-
lishing Corporation, 1980.
[27] F. Moukalled, L. Mangani, M. Darwish, The Finite Volume Method in
Computational Fluid Dynamics: An Advanced Introduction with Open-
FOAM and Matlab, Springer International Publishing, 2016.
685 [28] Ankur Garg, G Chanakya, Pradeep Kumar, Numerical error estimation in
finite volume method for radiative transfer equation for collimated irradi-
ation, in: Proceedings of the 9th International Symposium on Radiative
Transfer, RAD-19, June 3-7, 2019, Athens, Greece, 2019.
[29] Aswatha, C. J. Gangadhara Gowda, S. N. Sridhara, K. N. Seetharamu,
690 Effect of different thermal boundary conditions at bottom wall on natural
convection in cavities, J. of Engineering Science and Technology 6 (2011)
109 – 130.
[30] OpenFOAM, The open source cfd toolbox: User guide, openfoam v1706
(2017).
695 [31] Tanmay Basak, S. Roy, A.R. Balakrishnan, Effects of thermal boundary
conditions on natural convection flows within a square cavity, Int. J of
Heat and Mass Transfer 4525–4535 49 (2006) 4525–4535. doi:10.1016/j.
ijheatmasstransfer.2006.05.015.
44